Conservation of Momentum, Equilibrium of a Particle and friction

• When no external forces are acting on the system the total momentum of the system remains constant.
• Total momentum of system before collision is equal to the total momentum of system after collision.
• When two objects exert forces on each other in action reaction, their motions are affected as a pair  m₁v₁ + m₂v₂ = 0
• Recoil velocity of gun is \tt v_{g} = -\frac{-m_{b} v_{b}}{M_{g}}
• Change in momentum ΔP = mv – mu.
• Resultant momentum \tt \Delta P = \sqrt{P_{1}^{2} + P_{2}^{2} + 2P_{1}P_{2} \cos \theta}
• Change in momentum in opposite direction ΔP = m(v + u).
• A ball hits wall with “u” at an angle “θ” with normal to wall ΔP = 2 mu cos θ.
• Change in momentum along wall = 0.

Friction:

• The frictional force acting between two surfaces at rest with respect to each other is called static friction f_s = μ_s N.
• The frictional force between surfaces in motion is called Kinetic friction f_K = μ_K N.
• The frictional force when rolls over, the surface of another called Rolling friction f_R = μ_R N.
• The cofficient of friction  μ is  f_s > f_K > f_R       μ_s > μ_K > μ_{R .}
• The angle between net contact force and Normal reaction called angle of friction tan θ = fs/N.
• Acceleration of body moving over rough surface \tt a = \frac{F- \mu_k \mu_g}{M}
• Frictional force on a block of mass ‘M’ when a pushing force “F” is applied making an angle ‘θ’ with horizontal F = μ (μg − F sin θ)
• Minimum force acquired to pull \tt F = \frac{- \mu m g}{\cos \theta + \mu \sin \theta}
• Frictional force on a block of mass M when a pushing force ‘F’ is applied making an angle ‘θ’ with horizontal F = μ (μg + F sin θ)
• Minimum force required to pull \tt F = \frac{\mu m g}{\cos \theta - \mu \sin \theta}
• Acceleration of block pressed to vertical wall \tt a = \frac{Mg - f_{k}}{m}
• Block of mass ‘m’ kept on truck moving with a_τ (acceleration) acceleration of block \tt a_{B} = \frac{ma_{\tau} - f_{k}}{m}
• Velocity of block \tt v = \sqrt{2 (a_{\tau} - \mu k g) x}
• Time taken to move \tt t = \sqrt{\frac{2x}{\left(a_{\tau} - \mu k g\right)}}
• Acceleration of block of a smooth incline a = g sin θ
• Time taken to reach bottom t = \tt \sqrt{\frac{2 l}{g \sin \theta}}
• Velocity of block at bottom \tt v = \sqrt{2 g l \sin \theta}
• Acceleration of block on rough inclined plane a = g (sin θ – μK cos θ)
• Velocity of block to reach bottom \tt v = \sqrt{2 g l (\sin \theta - \mu k \cos \theta)}
• Time taken to reach bottom \tt t = \sqrt{\frac{2 l}{g (\sin \theta - \mu k \cos \theta)}}
• Force required to move the block up the plane F = mg (sin θ + μK cos θ).
• When the body thrown up with velocity “U” The retarding force F = mg (sin θ + μK cos θ).
• The retardation a = g (sin θ + μK cos θ)
• Time taken to stop \tt t = \sqrt{\frac{2 l}{g (\sin \theta + \mu k \cos \theta)}}
• Distance travelled by body to stop \tt s = \frac{u^{2}}{2 g (\sin \theta + \mu k \cos \theta)}
• Acceleration of system of two Blocks when Force is applied on Bottom block a_max = μs g.

•  Maximum force for which blocks move together F_max = μs (M+m)g
• Acceleration of Lower block \tt a = \frac{F - \mu k mg}{M}
• Acceleration of system of two blocks when force is applied on top block \tt a = \frac{\mu s mg}{M}

•  Maximum force for which both blocks move together \tt F = (M + m) \frac{\mu s mg}{M}
• Acceleration of lower block = \tt \frac{\mu_k mg}{M}
• Acceleration of upper block = \tt \frac{F - \mu_k mg}{m}